Reservoir modeling characterization is generally based on localized, and frequently sparse, observations that are interpolated to return spatial descriptions of uncertain geological properties, such as permeability. The interpolation process introduces uncertainty in the permeability field and translates directly into uncertainty about the reservoir behavior. Reconciling reservoir models with dynamic measurements and field production data, commonly referred to as history matching (HM), can reduce permeability uncertainty. In reservoir modeling, the impact of uncertainty on static geological models usually ranks as follows:
1. Structural Framework (defining gross volumes);
2. Stratigraphic layering (connectivity control);
3. Facies modeling (control over depositional continuity; and
4. Petrophysical modeling (property distribution).
In other words, realistic geological model representations should primarily define correct structural framework and account for the depositional continuity and connectivity control because these properties bare the most significant effect on the fluid flow within the reservoir. It is therefore, a common understanding that HM methods work best when they incorporate realistic facies information. When the permeability field is characterized by finely discretized block values of permeability, the HM problem can be ill-posed. Moreover, if estimated block values of permeability are not constrained to preserve facies connectivity, they may yield geologically inconsistent and unrealistic permeability fields. In order to control ill-posedness and to respect geological facies, a parametric distribution of permeability may be required that is low dimensional, which still preserves the important geological features and their connectivity. This requirement particularly holds true for geological structures with complex geometrical configurations, such as deltaic channels, fluvial deposits, turbidites and shale drapes.
Several conventional parametrization approaches have been proposed and implemented for addressing HM problems. Each approach represents the block values of permeability at any given location with a linear expansion composed of the weighted eigenvectors of the specified covariance matrix for block values of permeability. The computational expense of covariance matrix inversion for realistic field model conditions, increases dramatically when millions of parameters are required and/or when large numbers of geological models are generated. One recent attempt to avoid expensive covariance matrix computations is represented by the Fourier-filter based method as described by M. Maucec, et. al. in Streamline-based History Matching and Uncertainty: Markov-Chain Monte Carlo Study of an Offshore Turbidite Oil Field (SPE 109943), where the geological model updates are generated in the Fourier wave domain and an inverse Fast Fourier Transform (FFT) is used to convert back to model (permeability) space. The spatial correlation, however, is modeled with two-point geostatistics by defining a variogram, which makes the description of facies distribution highly challenging if not impossible. The upside of the method is definitely its speed: it is capable of generating a new realization of the permeability field with many variables (˜106) in just a few seconds, thus, omitting the computational and memory cost of the traditional approaches that represent decomposition algorithms.
Within the last decade, several advances have been made in the form of multi-point geostatistics (MPS), which uses correlations between multiple locations at the same time to reproduce volume-variance relationships and geological models conditioned to local sample data. Examples of MPS technology include the techniques described by S. Strebelle in Reservoir Modeling Using Multiple-Point Statistics (SPE 71324), and by N. Remy in A Geostatistical Earth Modeling Library and Software, a Stanford University thesis, which combine codes like SNESIM and S-GeMS, respectively. The last example is dedicated to the local optimization of parameters involved in variogram-based models to take into account local structural characteristics of the data. MPS, however, has a few main drawbacks, which include: i) dependence on the training images or training data sets; and ii) very long computational times for generating new geological models.
Moreover, current techniques that implement a Discrete Cosine Transform (“DCT”)-based parametrization in HM workflows only addresses small-size, two-dimensional (2D) problems with bi-modal permeability distribution.